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        <h1 id="数学"><a href="#数学" class="headerlink" title="数学"></a>数学</h1><h2 id="1"><a href="#1" class="headerlink" title="1"></a>1</h2><h3 id="极限"><a href="#极限" class="headerlink" title="极限"></a>极限</h3><ul>
<li><p>核心考点</p>
<ul>
<li>定义</li>
<li>性质</li>
<li>计算</li>
<li>应用</li>
</ul>
</li>
<li><p>定义</p>
<ul>
<li></li>
</ul>
</li>
</ul>
<h2 id="2"><a href="#2" class="headerlink" title="2"></a>2</h2><h2 id="3"><a href="#3" class="headerlink" title="3"></a>3</h2><h2 id="4"><a href="#4" class="headerlink" title="4"></a>4</h2><h2 id="6"><a href="#6" class="headerlink" title="6"></a>6</h2><h3 id="微分方程"><a href="#微分方程" class="headerlink" title="微分方程"></a>微分方程</h3><ul>
<li><p>计算</p>
<ul>
<li><p>一阶微分方程的求解</p>
<ul>
<li></li>
<li><p>子主题 2</p>
<ul>
<li>子主题 3</li>
</ul>
</li>
<li><p>子主题 5</p>
</li>
<li>子主题 6</li>
<li>子主题 5</li>
</ul>
</li>
<li><p>二阶可降解微分方程</p>
</li>
<li>高阶常系数线性微分方程</li>
<li><p>换元法求解微分方程</p>
<ul>
<li>求导公式逆用换元</li>
<li>自变量换元</li>
<li>因变量</li>
<li>xy地位互换换元</li>
</ul>
</li>
</ul>
</li>
<li><p>应用</p>
<ul>
<li>用极限导数定义建方程</li>
<li><p>用几何应用建方程</p>
<ul>
<li>曲线切线斜率</li>
<li>f1f2的公切线</li>
<li>截距</li>
<li>面积体积平均值弧长侧面积曲率质心</li>
</ul>
</li>
<li><p>用变化建方程</p>
<ul>
<li>元素衰变问题</li>
<li>人口增长</li>
<li>物线追踪</li>
<li>冷却问题</li>
<li>牛二</li>
</ul>
</li>
</ul>
</li>
</ul>
<h2 id="10"><a href="#10" class="headerlink" title="10"></a>10</h2><h2 id="分支主题-17"><a href="#分支主题-17" class="headerlink" title="分支主题 17"></a>分支主题 17</h2><h3 id="一元函数积分"><a href="#一元函数积分" class="headerlink" title="一元函数积分"></a>一元函数积分</h3><h3 id="几何应用"><a href="#几何应用" class="headerlink" title="几何应用"></a>几何应用</h3><h3 id="研究对象"><a href="#研究对象" class="headerlink" title="研究对象"></a>研究对象</h3><ul>
<li>f(x)</li>
<li><p>fn(x)</p>
<ul>
<li>x=x(t)</li>
<li>y=y(t)</li>
</ul>
</li>
<li></li>
<li></li>
<li>微分方程的解f(x)</li>
<li>面积</li>
<li>旋转体体积</li>
<li>子主题 9</li>
<li>平均值</li>
<li>平面曲线的弧长</li>
<li>旋转曲面的面积</li>
<li>平面上的曲边梯形</li>
<li>平行截面面积为已知的立体图形</li>
</ul>
<h3 id="研究内容"><a href="#研究内容" class="headerlink" title="研究内容"></a>研究内容</h3><h2 id="15"><a href="#15" class="headerlink" title="15"></a>15</h2><h3 id="微分方程-1"><a href="#微分方程-1" class="headerlink" title="微分方程"></a>微分方程</h3><ul>
<li>一阶微分方程的求解</li>
</ul>
<h2 id="16"><a href="#16" class="headerlink" title="16"></a>16</h2><h3 id="无穷级数"><a href="#无穷级数" class="headerlink" title="无穷级数"></a>无穷级数</h3><ul>
<li><p>数列级数的判敛</p>
<ul>
<li>定义 Sn</li>
<li><p>判敛法</p>
<ul>
<li>正向级数</li>
<li>交错级数</li>
<li>任意项级数</li>
</ul>
</li>
<li><p>常用结论</p>
</li>
</ul>
</li>
<li><p>幂级数的收敛域</p>
<ul>
<li><p>概念</p>
<ul>
<li>幂级数</li>
<li>收敛点与发散点</li>
</ul>
</li>
<li><p>具体性问题</p>
<ul>
<li>anxn</li>
<li>缺项性问题</li>
</ul>
</li>
<li><p>抽象性问题</p>
<ul>
<li>阿贝尔定理</li>
<li>结论1</li>
<li>结论2</li>
</ul>
</li>
</ul>
</li>
<li><p>展开问题</p>
<ul>
<li><p>考法</p>
<ul>
<li>函数展开</li>
<li>积分展开</li>
<li>导数展开</li>
<li>无穷小比阶</li>
</ul>
</li>
<li><p>攻工具</p>
<ul>
<li>先积后导</li>
<li>先导厚积</li>
<li>重要展开公式</li>
</ul>
</li>
</ul>
</li>
<li><p>求和问题</p>
<ul>
<li>直接套公式</li>
<li>用先积后导或先导厚积求和函数</li>
<li>用所给微分方程求和函数</li>
<li>建立微分方程并求和函数</li>
<li>综合题</li>
</ul>
</li>
<li><p>傅里叶级数</p>
<ul>
<li>迪利克雷收敛</li>
<li>周期为2I的周期函数的傅里叶级数与系数公式</li>
</ul>
</li>
</ul>
<h2 id="分支主题-15"><a href="#分支主题-15" class="headerlink" title="分支主题 15"></a>分支主题 15</h2><h2 id="分支主题-13"><a href="#分支主题-13" class="headerlink" title="分支主题 13"></a>分支主题 13</h2><h2 id="分支主题-14"><a href="#分支主题-14" class="headerlink" title="分支主题 14"></a>分支主题 14</h2><h2 id="19"><a href="#19" class="headerlink" title="19"></a>19</h2><h3 id="行列式"><a href="#行列式" class="headerlink" title="行列式"></a>行列式</h3><ul>
<li><p>定义与性质</p>
<ul>
<li><p>n阶行列式定义</p>
<ul>
<li><p>n个向量为邻边的n维图形的n维体积</p>
<ul>
<li>子主题 1</li>
</ul>
</li>
</ul>
</li>
<li><p>性质</p>
<ul>
<li>行列互换、其值不变</li>
<li>行列式中某行（列）元素全为零，则行列式为零</li>
<li>行列式中两行（列）元素相等或对应成比例，则行列式为零</li>
<li>行列式中某行元素均是两个元素之和，则可以拆成两个行列式之和</li>
<li>行列式中俩行互换，值变号</li>
<li>行列式中某行有公因子k，则k可以外放</li>
<li>行列式种某行的k倍加到零一行，值不变</li>
</ul>
</li>
<li><p>行列式展开定理</p>
<ul>
<li>余子式</li>
<li>代数余子式</li>
</ul>
</li>
</ul>
</li>
<li><p>具体型行列式的计算</p>
<ul>
<li><p>化为12+1型行列式</p>
<ul>
<li>主对角线</li>
<li>副对角线</li>
<li>拉普拉斯展开式</li>
<li>范德蒙德行列式</li>
</ul>
</li>
<li><p>加边法</p>
</li>
<li><p>递推法</p>
<ul>
<li>找出递推公式，Dn与Dn+1</li>
</ul>
</li>
<li><p>数学归纳法</p>
</li>
</ul>
</li>
<li><p>抽象性行列式的计算</p>
<ul>
<li>用行列式性质</li>
<li><p>用矩阵知识</p>
<ul>
<li>设C=AB，|C|=|AB|=|A||B|</li>
<li>设C=A+B，|C|=|A+B| 做恒等变换</li>
<li>设A，则|A*|=|A|^n-1，</li>
<li>若A相似于B，则|A|=|B|</li>
</ul>
</li>
<li><p>用方程知识</p>
</li>
</ul>
</li>
</ul>
<h2 id="28"><a href="#28" class="headerlink" title="28"></a>28</h2><h3 id="随机事件和概率"><a href="#随机事件和概率" class="headerlink" title="随机事件和概率"></a>随机事件和概率</h3><ul>
<li><p>古典概型求概率</p>
<ul>
<li><p>随机分配问题</p>
<ul>
<li>每盒容纳任意多个质点</li>
<li>每盒容纳至多1个质点</li>
</ul>
</li>
<li><p>简单随机抽样问题</p>
<ul>
<li>先后有放回</li>
<li>先后无放回</li>
<li>任取</li>
</ul>
</li>
</ul>
</li>
<li><p>几何概型求概率</p>
<ul>
<li>P(A)=A的度量/M的度量</li>
</ul>
</li>
<li><p>重要公式求概率</p>
<ul>
<li>互斥</li>
<li>对立</li>
<li>独立</li>
<li>条件</li>
<li>不等式</li>
<li>最值</li>
</ul>
</li>
<li><p>事件独立性</p>
<ul>
<li><p>定义</p>
<ul>
<li>P(AB)=P(A)P(B)</li>
</ul>
</li>
<li><p>判定</p>
</li>
</ul>
</li>
</ul>
<h2 id="29"><a href="#29" class="headerlink" title="29"></a>29</h2><h3 id="一维随机变量及其分布"><a href="#一维随机变量及其分布" class="headerlink" title="一维随机变量及其分布"></a>一维随机变量及其分布</h3><ul>
<li><p>判分布</p>
<ul>
<li><p>随机变量及其分布函数的定义</p>
<ul>
<li>随机变量</li>
<li>分布函数</li>
</ul>
</li>
<li><p>判分布</p>
<ul>
<li><p>分布函数</p>
<ul>
<li>F(x)单调不减</li>
<li>F(x)右连续</li>
<li>F(负无穷)=0</li>
<li>F(正无穷)=1</li>
</ul>
</li>
<li><p>概率密度</p>
<ul>
<li>pi&gt;=0且Σpi=1</li>
<li><p>∫f(x)dx=1</p>
<ul>
<li><ul>
<li></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li><p>反问题</p>
</li>
</ul>
</li>
</ul>
</li>
<li><p>求分布</p>
<ul>
<li><p>离散型分布</p>
<ul>
<li>0-1分布</li>
<li>二项分布</li>
<li>几何分布</li>
<li>超几何分布</li>
<li>泊松分布</li>
</ul>
</li>
<li><p>连续型分布</p>
<ul>
<li>均匀分布</li>
<li>指数分布</li>
<li>正态分布</li>
</ul>
</li>
<li><p>混合型</p>
<ul>
<li>F(x)=P{X&lt;=x}</li>
</ul>
</li>
<li><p>实际问题</p>
<ul>
<li><p>已知离散求离散</p>
<ul>
<li></li>
</ul>
</li>
<li><p>已知连续求连续</p>
<ul>
<li></li>
</ul>
</li>
<li><p>已知连续求离散</p>
<ul>
<li></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li><p>用分布</p>
<ul>
<li></li>
<li></li>
<li></li>
<li>反问题</li>
</ul>
</li>
</ul>
<h2 id="30"><a href="#30" class="headerlink" title="30"></a>30</h2><h3 id="多维随机变量及其分布"><a href="#多维随机变量及其分布" class="headerlink" title="多维随机变量及其分布"></a>多维随机变量及其分布</h3><h2 id="数字特征"><a href="#数字特征" class="headerlink" title="数字特征"></a>数字特征</h2><h3 id="期望"><a href="#期望" class="headerlink" title="期望"></a>期望</h3><ul>
<li><p>X</p>
<ul>
<li>x~Pi =&gt; EX=Σxp</li>
<li>x~f(x) =&gt; EX=∫xf(x)dx</li>
<li>无穷级数</li>
<li>有限</li>
<li>定积分</li>
<li>反常积分</li>
<li>子主题 7</li>
</ul>
</li>
<li><p>g(x)</p>
</li>
<li>g(x,y)</li>
<li>最值</li>
<li>分解</li>
<li>性质</li>
</ul>
<h3 id="方差"><a href="#方差" class="headerlink" title="方差"></a>方差</h3><ul>
<li><p>X</p>
<ul>
<li><p>定义法</p>
<ul>
<li>DX=E(X-EX)²</li>
<li>DX=Σ(x-EX)²Pi</li>
<li>DX=∫(x-EX)²f(x)dx</li>
</ul>
</li>
<li><p>公式法</p>
<ul>
<li>DX=EX²-(EX)²</li>
</ul>
</li>
</ul>
</li>
<li><p>最值</p>
<ul>
<li>DX=∫x²f(x)dx</li>
</ul>
</li>
<li><p>分解</p>
<ul>
<li>DX=DX1+DX2+……+DXn+2Σcov(xi,xj)</li>
</ul>
</li>
<li><p>性质</p>
<ul>
<li>DX≮0</li>
<li>Dc=0</li>
<li>P{X=a}=1</li>
<li>D(ax+b)=a²Dx</li>
<li>D(X±Y)=DX+DY±2COV(X,Y)</li>
<li><p>X,Y独立</p>
<ul>
<li>D(aX+bY)=a²DX+B²DY</li>
<li>D(XY)=DXDY+DX(EX)²+DY(DX)²&gt;=DXDY</li>
</ul>
</li>
<li><p>DX=E[{X-EX)²]&lt;=E[(X-c)²]</p>
</li>
</ul>
</li>
</ul>
<h3 id="常用EXDX"><a href="#常用EXDX" class="headerlink" title="常用EXDX"></a>常用EXDX</h3><ul>
<li><p>0-1分布</p>
<ul>
<li><p>p</p>
<ul>
<li>pq</li>
</ul>
</li>
</ul>
</li>
<li><p>X~B(n,p)</p>
<ul>
<li><p>np</p>
<ul>
<li>npq</li>
</ul>
</li>
</ul>
</li>
<li><p>X~P(λ)</p>
<ul>
<li><p>λ</p>
<ul>
<li>λ</li>
</ul>
</li>
</ul>
</li>
<li><p>X~Ge(p)</p>
<ul>
<li><p>1/p</p>
<ul>
<li>（1-p)/p²</li>
</ul>
</li>
</ul>
</li>
<li><p>X~V(a,b)</p>
<ul>
<li><p>(a+b)/2</p>
<ul>
<li>(b-a)²/12</li>
</ul>
</li>
</ul>
</li>
<li><p>X~E(λ)</p>
<ul>
<li><p>1/λ</p>
<ul>
<li>1/λ²</li>
</ul>
</li>
</ul>
</li>
<li><p>X~N(μ,σ)</p>
<ul>
<li><p>μ</p>
<ul>
<li>σ²</li>
</ul>
</li>
</ul>
</li>
<li><p>X~X²(n)</p>
<ul>
<li><p>n</p>
<ul>
<li>2n</li>
</ul>
</li>
</ul>
</li>
</ul>
<h3 id="cov与pxy"><a href="#cov与pxy" class="headerlink" title="cov与pxy"></a>cov与pxy</h3><ul>
<li><p>cov(x,y)</p>
<ul>
<li>定义</li>
<li>定义法</li>
<li>公式法</li>
</ul>
</li>
<li><p>pxy</p>
</li>
<li>性质</li>
</ul>
<h3 id="独立性与相关性判断"><a href="#独立性与相关性判断" class="headerlink" title="独立性与相关性判断"></a>独立性与相关性判断</h3><ul>
<li>用分布判分布</li>
<li>用数字特征判分布</li>
<li>程序</li>
<li>重要结论</li>
</ul>
<h3 id="切比雪夫不等式"><a href="#切比雪夫不等式" class="headerlink" title="切比雪夫不等式"></a>切比雪夫不等式</h3><h3 id=""><a href="#" class="headerlink" title=" "></a> </h3><ul>
<li></li>
<li></li>
<li></li>
<li>Γ（1）=1</li>
</ul>
<h2 id="判分布"><a href="#判分布" class="headerlink" title="判分布"></a>判分布</h2><h3 id="F-x-y-是联合分布函数的充要条件"><a href="#F-x-y-是联合分布函数的充要条件" class="headerlink" title="F(x,y)是联合分布函数的充要条件"></a>F(x,y)是联合分布函数的充要条件</h3><ul>
<li>单调不减函数</li>
<li>右连续性</li>
<li>有界性</li>
<li>非负</li>
</ul>
<h3 id="P0是联合分布律的充要条件"><a href="#P0是联合分布律的充要条件" class="headerlink" title="P0是联合分布律的充要条件"></a>P0是联合分布律的充要条件</h3><ul>
<li>p&gt;=0且ΣΣp=1</li>
</ul>
<h3 id="f-x-y-是联合概率密度的充要条件"><a href="#f-x-y-是联合概率密度的充要条件" class="headerlink" title="f(x,y)是联合概率密度的充要条件"></a>f(x,y)是联合概率密度的充要条件</h3><ul>
<li>f(x,y)&gt;=0且∫∫=1</li>
</ul>
<h3 id="反问题"><a href="#反问题" class="headerlink" title="反问题"></a>反问题</h3><h2 id="求分布"><a href="#求分布" class="headerlink" title="求分布"></a>求分布</h2><h3 id="求联合分布"><a href="#求联合分布" class="headerlink" title="求联合分布"></a>求联合分布</h3><ul>
<li>求F(x,y)</li>
<li>求Pij</li>
<li>求f(x,y)</li>
</ul>
<h2 id="分支主题-2"><a href="#分支主题-2" class="headerlink" title="分支主题 2"></a>分支主题 2</h2>
      
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